1. Introduction

The generation of terahertz sources through air-excited filament has garnered significant attention due to its exceptional performance and broad applications in fields such as terahertz spectroscopy, imaging technology, and terahertz-matter interactions [1–3]. In 2000, Cook et al. demonstrated the production of robust terahertz waves through air filament excitation using fundamental and second harmonic (SH) two-color pulses and introduced a four-wave mixing (FWM) model to elucidate this phenomenon [4], which attracted many researchers to further investigate the underlying physical mechanism [5–10].

In 2007, based on semi-classical theory, K. Y. Kim, T. I. Oh et al proposed a transient photocurrent model to explain the terahertz generation from plasma filament induced by two-color field [9], in which they attributed terahertz generation to the asymmetric photocurrent induced by the uneven two-color field and caused wide attention. Subsequently, they further did a lot works based on this model and made significant progresses [11–13]. In 2008, M. Chen et al adopted particle-in-cell (PIC) method to numerically simulate the process based on transient photocurrent model [14,15], by which the propagation effect of this process is effectively taken into account. However the PIC method is computationally very expensive and only fewer micrometers space can be simulated. To overcome the trouble of expensive computations, based on the transient photocurrent model, S. Skupin et al developed it to plasma current form and incorporate the plasma current into Maxwell’s equations to self-consistently generate terahertz emission under macroscopic situation [16], and adopted the unidirectional pulse propagation equation (UPPE) [17] method to compute the forward terahertz emission. Furthermore, in order to calculate the backward terahertz emission, C. Köhler et al adopted the FDTD method to simulate terahertz generation by including the plasma current into the Maxwell’s equations, which self-consistently gave both the forward and backward terahertz emissions, and triumphantly explained the forward and backward terahertz emission via superposition effect of interfering signals [18].

To differentiate the contributions from different mechanisms, in 2016, O. G. Kosareva et al theoretically and experimentally investigated the terahertz emission from plasma filament and confirmed that photocurrent plays a predominant role in low-frequency range whereas the four-wave mixing and wakefield oscilation play a negligible role [19]. Since then, A. A. Ushakov et al further experimentally studied the waveform, spectrum, and energy of back terahertz emission from plasma filament excited by a two-color femtosecond laser under different pump pulse energy and obtained results that quantitatively agree with numerical simulations [20,21]. Moreover, Y. Chen, Z. Tian, J. Dai et al investigated the dependence of peak terahertz amplitude on the laser pulse energy and revealed that the forward and backward terahertz generation processes are essentially the same [22].

Besides, many other related topics have also been extensively studied. For instance, Y. Bai et al demonstrated a novel scheme to generate waveform-controlled terahertz emission from air plasma produced when carrier-envelope-phase (CEP) stabilized few-cycle laser pulses undergo filamentation in ambient air [23]. T. J. Wang et al reported that the longitudinal evolution of amplified terahertz emission has been demonstrated by applying a needlelike high-voltage direct current (DC) electric field on the laser filament [24]. In addition, S. Zhang et al delved into the effects of three-color laser pulses and pulse duration on terahertz generation [25,26], Y. Peng et al extended the model to accommodate fluctuations in local gas density [27].

Despite the significant progresses the photocurrent model has achieved in various aspects, the more substantial problem about whether/how the plasma-current-emitted field (it is called photocurrent radiation as follows) affects the terahertz generation in the entire physical process is still unclear, which allows us to conduct a systematic research to provide a reasonable conclusion to it.

In this paper, we revisit the fundamental mechanism responsible for terahertz generation from laser-induced plasma filament based on the photocurrent model by employing a blend of analytical calculation and numerical simulation. By taking full advantage of the FD-FDTD method [28–30], the roles of two-color field and photocurrent radiation in terahertz generation from plasma filament are visually separated, and the driving effect of photocurrent radiation is confirmed pretty significant. Furthermore, through careful analysis and computation, it is revealed that plasma-induced modulation to photocurrent radiation is actually the underlying physical mechanism of terahertz generation from plasma filament. Moreover, a three-step diagram is introduced to reillustrate the overall physical process and provides a more comprehensive explanation than the ever work that utilizes superposition effect of interfering signals alone [18].

This paper commences by presenting a specific theoretical model for terahertz generation from plasma filament, including the decomposed Maxwell’s equations employed in the FD-FDTD method, plasma current formula, and the other related equations. Subsequently, the main conditions adopted in this work are declared and the numerical simulation process is also specified in detail. Then, an argon gas point excited by two-color field is adopted to confirm the validity and correctness of the FD-FDTD method when simulating terahertz generation from plasma filament, through comparing the numerical result of photocurrent radiations with an accurate result derived from one-dimensional radiation formula. Through artificially removing or introducing the photocurrent radiation term in plasma current equation when using FD-FDTD method to simulate the terahertz generation process, a pair of totally different results are obtained, so the driving effect of photocurrent radiation is confirmed to be significant in terahertz generation. Subsequently, through further investigating the driving effect of photocurrent radiation, the underlying mechanism of terahertz emission from laser-induced plasma filament is proven to be plasma-induced modulation to photocurrent radiation instead of the establishment of bias voltage on the plasma, and a three-step diagram is proposed to explain the entire physical process. In addition, the plasma-induced modulation mechanism for terahertz generation is substantiated by taking a theoretical prediction of the minimal filament length required for achieving stable backward terahertz emission and obtaining consistent results with the numerical simulations.

2. Theoretical model

As a linearly polarized intense laser pulse propagates in a gaseous medium, its temporal evolution is governed by the Maxwell’s equations together with field-ionized model, such as tunnel ionization, etc. When the propagation distance is very short (far less than the Rayleigh length), the transverse diffraction and self-focusing effects can be ignored, which is helpful for highlighting the driving effect of photocurrent radiation on terahertz generation. Henceforth, the entire process is going to be expressed by one-dimensional Maxwell’s equations as,

Based on the transient photocurrent model proposed by K. Y. Kim et al [12], when the initial velocity of newly born electrons is assumed zero and the ponderomotive force is neglected, the microscopic transient photocurrent $J(t) = q\int _{-\infty }^{t} v(t,t^{\prime }) d\rho _\text {e}(t^{\prime })$ reduces to macroscopic plasma current [16]:

For more details about the derivation process, please refer to the Appendix A. Here, $m_\text {e}$ and $e$ respectively represent the static mass and charge of the electron. The electric field $E_\text {x}$ is determined by Eq. (1), by which the two equations are coupled together. In addition, $\tau _\text {c}$ is the average collision time of electrons with other particles, and $\rho _\text {e}$ is the free electron density which is governed by

It should be noted that the electric field in Eq. (1) has mixed together the pump field and the plasma-current-emitted photocurrent radiation, which makes it unable to synchronously separate the electric field into two parts (i. e., the pump field and the photocurrent radiation) to respectively investigate the role of each field during simulation. However, studying the role of each electric field in generating terahertz waves from laser-induced filament is very helpful to further understand the physical mechanism behind it. In order to achieve this end, we decompose Eq. (1) into two Maxwell’s sub-equations and adopt the FD-FDTD method to simulate the whole dynamic process. The first equation governs the behavior of the pump field $E_{\text {x,p}}$,

3. Numerical method

In this work, the pump pulse is assumed as a linearly-polarized two-color laser consisting of 800-nm field ($\nu _\text {f} \sim 375$ THz) and its second harmonic wave (400 nm, $\nu _\text {s} \sim 750$ THz) formulated by

When the pump field and photocurrent radiation propagates within the 40-microns argon gas, the phase difference caused by dispersion is very small so that the dispersion can be ignored [33]. Thus, the polarizations in Eq. (5) and Eq. (6) can be presented by $P_{\text {x,p}} = \varepsilon _0 \chi _{\text {p}} E_{\text {x,p}}$ and $P_{\text {x,r}} = \varepsilon _0 \chi _{\text {r}} E_{\text {x,r}}$ with the susceptibility $\chi _{\text {p}} = \chi _{\text {r}} = 5.56 \times 10^{-4}$. Moreover, assuming the transverse size of filament and pump field is $d$, the final electron density is $\rho _\text {e}(+\infty ) \simeq 2 \times 10^{24}$ m$^{-3}$, the intensity of pump field is $I = c\varepsilon _0 n E_0^2 / 2 \simeq 128$ TW/cm$^{2}$, the temporal width of pump field is $T = 2\tau \simeq 70$fs, and the length of filament is $L = 40\,\mu$m. Thus, the energy of pump field is $W_{\text {laser}} = I T d^2$ and the energy of ionization is $W_{\text {i}} = U_{\text {i}} \rho _\text {e}(+\infty ) d^2 L$, and then the energy ratio is defined as $R \equiv W_{\text {i}} / W_{\text {laser}} = I T / [U_\text {i} L \rho _\text {e}(+\infty )] \simeq 0.2{\% }$, which means the energy loss is very small and is negligible for the analysis. Thus, the loss current $J_{\text {loss}}$ related to photo-ionization attenuation is disregarded in Eq. (5). What’s more, the loss of pump field for driving ionized electrons is contained in the plasma current $J_\text {e}$. In addition, the terahertz waves generated from the FWM effect and wakefield oscillation of plasma are ignored because the contribution of these two phenomena to terahertz emission is very limited in contrast to photocurrent radiation [19], especially in low-frequency range.

Based on the above conditions, we now adopt the FD-FDTD method to simulate this process by inputting a two-color pump pulse through a 40-$\mu$m-thick argon gas to generate terahertz emission. First, Eq. (5) is used to calculate the spatiotemporal evolution of the pump field $E_{\text {in}}$ of Eq. (9) in the medium. Second, Eq. (3) and Eq. (7) are used to calculate the electron density $\rho _\text {e}$ and plasma current $J_\text {e}$, respectively. Third, Eq. (6) is used to calculate the generation and propagation of photocurrent radiation $E_{\text {x,r}}$ in filament. Fourth, repeat the first three steps again and again until all time steps are completed . Fifth, the terahertz wave is obtained by filtering out the components above 100 THz in photocurrent radiation recorded in front or behind the filament. Through comparing the terahertz waveform and spectrum obtained under situations of 1) considering and 2) ignoring the photocurrent radiation $E_{\text {x,r}}$ in Eq. (7), the impact of driving effect of photocurrent radiation on terahertz generation can be quantitatively determined, which cannot be realized using the traditional FDTD method because the pump field $E_{\text {x,p}}$ and the photocurrent radiation $E_{\text {x,r}}$ are combined into field $E_\text {x}$.

4. Algorithm verification

While the robustness of the FD-FDTD method has been extensively proved, it is necessary to further conduct a verification to be on the safe side. As an illustrative example, we employ this method to compute the photocurrent radiation emitted from a 2 nm argon gas spot excited by two-color laser given by Eq. (9) via numerically solving Eqs. (3), (5–7). As a contrast, the accurate results of photocurrent radiation can be obtained via one-dimensional radiation formula. The complete derivation of one-dimensional radiation formula is provided in Appendix C. Now, we assume the gas point centers at $z = 0$ nm ($-1\sim 1$ nm), and the plasma current $J_\text {e}$ is almost unchanged within the 2 nm range. Thus, the photocurrent radiation $E_{\text {r}}$ calculated according to the one-dimensional radiation formula at point $z$ obeys by

Figure 1 illustrates the waveforms and spectra derived from the two approaches. First, Fig. 1(a) shows a perfect overlap for the temporal profiles, exhibiting a pronounced initial surge followed by a gradual decline. This temporal profile is what we called photocurrent radiation. Second, Fig. 1(b) displays a substantially consistent spectra in logarithm-scale axes with an error on $10^{-5}$ magnitudes, which can be considered completely equivalent. Thus, we can conclude the Fig. 1 affirms the accuracy of the FD-FDTD method in treatment of terahertz generation from laser-induced plasma filament excited by two-color field.

 

Fig. 1. Temporal profiles and spectra of the photocurrent radiation emitted from a 2-nm-length argon gas spot. The results are respectively obtained by the FD-FDTD method (red-solid line) and the one-dimensional radiation formula (black-circle line): (a) depicts the temporal profiles corresponding to 0 - 100 THz frequency range; (b) illustrates the spectra of normalized photocurrent radiation on logarithmic scale.

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5. Numerical results

Based on the above proof, we proceed to employ the FD-FDTD method with confidence to investigate whether the driving effect of photocurrent radiation holds a significant impact on terahertz generation from plasma filament excited by two-color pump pulse. In this simulation, we adopt the pump field provided by Eq. (9) to excite a 40-$\mu$m argon gas layer. By artificially introducing or removing the photocurrent radiation $E_{\text {x,r}}$ in the Eq. (7), the impact of driving effect of photocurrent radiation on terahertz generation can be visually demonstrated.

Figure 2 portrays two markedly distinct outcomes, involving both the temporal profiles and spectra of forward and backward terahertz emissions, represented by red-dashed line (removing $E_{\text {x,r}}$) and black-solid line (introducing $E_{\text {x,r}}$). The characteristic of spectra is presented in Figs. 2(a) and 2(c). The red-dashed line predominantly concentrates on the frequency range of $0 \sim 5$THz with a peak at the zero frequency point, which contradicts sharply with the spectra of black-solid line that exhibits a discernible band width. Furthermore, Figs. 2(b) and 2(d) illustrate the temporal profiles of terahertz emission under situations of removing and introducing photocurrent radiation respectively, which also presents substantial disparities. The pronounced dissimilarity between the two cases, as evidenced by Figs. 2(a-d), confirms the decisive impact of driving effect of photocurrent radiation on terahertz generation from plasma filament.

 

Fig. 2. The terahertz emissions in the situations of removing (red-dashed line) or introducing (black-solid line) the impact of photocurrent radiation $E_{\text {x,r}}$: (a) and (c) are the backward and forward spectra respectively and they are normalized by their peak values; (b) and (d) are the corresponding backward and forward temporal profiles.

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In addition, to elucidate why the elimination of driving effect of photocurrent radiation yields such a significant contrast is very helpful for further comprehending the underlying physical mechanism of terahertz generation.

In Fig. 2(d), it is very evident that the red-dashed line bears a striking resemblance to Fig. 1(a) except the difference in magnitude. Essentially, the red-dashed line in Fig. 2(d) can be obtained by scaling the temporal profile of Fig. 1(a) with a constant factor. This result suggests that disregarding the driving effect of photocurrent radiation leads to the following phenomena: 1) every point generates, in both directions, the same photocurrent radiation with temporal profile depicted in Fig. 1(a) because they are excited by nearly identical excitation pulse; 2) the photocurrent radiations generated at different point attain same phase at the forward observation point because the excitation field and the generated photocurrent radiations have the same velocity. Therefore, the photocurrent radiations generated at different points engage in constructive interference at the forward observation point, culminating in the emergence of the red-dashed line exhibited in Fig. 2(d).

In Fig. 2(b), a distinguished temporal profile from the Fig. 2(d) in the red-dashed line is obtained, characterized by a gradual enhancement followed by slow decay. In the situation of backward emission, the photocurrent radiations generated at different points experience different phase shift, resulting in a gentle amplification instead of a rapid upsurge, as shown in Fig. 2(d). To explain this phenomenon, a convincing schematic diagram is very necessary, which is depicted in Fig. 3.

 

Fig. 3. The schematic diagram of backward photocurrent radiations from the first emission point $P_\text {f}$ and the last emission point $P_\text {l}$ under situation of disregarding the driving effect of photocurrent radiation in Eq. (7). The photocurrent radiations emitted at points $P_\text {f}$ and $P_\text {l}$ have a time shift $2L/c$ when propagating to the backward observation point $P_{\text {BW}}$.

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Here, the first emission point $P_\text {f}$ and the last emission point $P_\text {l}$ are separated by a finite distance $L$. This distance leads to a time shift $2L/c$ for the photocurrent radiations emitted at points $P_\text {f}$ and $P_\text {l}$ at the backward observation point $P_{\text {BW}}$. Notably, this time shift corresponds to twofold distance between the two points. Thus, the generation process of backward photocurrent radiation can be explained as: 1) the pump field $E_{\text {x,p}}$ traverses from the first point $P_\text {f}$ through to the last point $P_\text {l}$; 2) the emitted photocurrent radiation $E_{\text {x,r}}$ at point $P_\text {l}$ retraces the path again to reach the backward observation point $P_{\text {BW}}$ via point $P_\text {f}$. Based on this explanation, when all backward photocurrent radiations converge at the backward observation point $P_{\text {BW}}$, their gradual enhancement eventually transforms into attenuation due to the absence of newly-generated backward photocurrent radiation advancing towards the backward observation point $P_{\text {BW}}$.

Through the preceding simulation and analysis, the manifestation of the red-dashed line can be explained by the superposition effect of interfering signals alone. Put differently, the intricate outcomes depicted by the black-solid line contradicts with the straightforward explanation of superposition mechanism, which means that there must be a more complicated physical process behind it. Therefore, the divergence between black-solid line and red-dashed line in Fig. 2 uncovers the significance of driving effect of photocurrent radiation on terahertz generation from plasma filament.

6. Mechanism analysis

Although it is confirmed that the driving effect of photocurrent radiation does have significant impact on terahertz generation from plasma filament, the underlying physical mechanism of the driving effect remains unclear because there still exits two potential mechanisms. One possible mechanism is photocurrent radiation builds up a bias voltage on the argon gas to promote ionization and to enhance the photocurrent asymmetry, which is equivalent to exert a couple of electrode plates on both sides of the argon gas, as depicted in Fig. 4(a). The another potential mechanism is the photocurrent radiation generated by two-color field experiences the plasma-induced modulation, which introduces dispersion and loss to the photocurrent radiations generated at different points in different degree, as illustrated in Fig. 4(b). In order to determine which one is the right physical mechanism of driving effect of photocurrent radiation, two well-designed numerical experiments are conducted.

 

Fig. 4. The schematic diagram of two potential mechanisms: (a) the two-color field $E_{\text {in}}$ together with photocurrent radiation $E_\text {r}$ ionizes a 23.5-$\mu$m argon gas to construct plasma filament, and then the ionized electrons are accelerated by the two fields to generate terahertz emission; (b) the photocurrent radiation $E_\text {r}$ generated by a 4-nm argon gas spot experiences the modulation induced by a $23.5$-$\mu$m uniform plasma, and then generates terahertz emission.

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We start by analyzing the mechanism illustrated in Fig. 4(a). First, we input the biased two-color field $E^{\prime }_{\text {in}}(t) = E_{\text {in}}(t) + E_{\text {r}}(t)$ to excite a 23.5 $\mu$m argon gas and construct a plasma filament. Here, $E_{\text {in}}$ is given by the Eq. (9) and the photocurrent radiation $E_{\text {r}}(t)$, namely the temporal profile depicted in Fig. 1(a), can be obtained according to Eq. (10). Second, the photocurrent radiation $E_{\text {r}}$ enhances the asymmetry of plasma current via establishing bias voltage on the plasma filament. Third, the plasma filament emits towards both sides, and the backward (toward left) emission is observed at point $P_{\text {BW}}$. In this case, we hypothesize the biased two-color field $E_{\text {in}}^{\prime }$ act as excitation field to ionize argon gas and then to generate plasma current $J_\text {e}$ via Eq. (7). Moreover, it should be noted that the synchronously-generated photocurrent radiation $E_{\text {x,r}}$ is not involved into the Eq. (7). Through resolving the Eqs. (3,5–7) using the above hypothesis, the photocurrent radiation is depicted by the red-dashed line in Fig. 5, which is almost consistent with the red-dashed line in Fig. 2. Therefore, this result implies that the interpretation of photocurrent radiation serving as a bias voltage is incorrect.

 

Fig. 5. The simulation results of backward terahertz emission under two potential mechanisms: (a) backward spectra and (b) backward temporal profiles. The red-dashed line depicts the results that two-color field $E_{\text {in}}$ together with photocurrent radiation $E_\text {r}$ excites a 23.5-$\mu$m argon gas. The black-solid line depicts the results that photocurrent radiation $E_{\text {r}}$ emitted by two-color field $E_{\text {in}}$ exciting a 4-nm argon gas spot experiences modulation induced by a 23.5-$\mu$m uniform plasma.

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Subsequently, we proceed to analyze the mechanism depicted in Fig. 4(b). First, the two-color field $E_{\text {in}}$ given by Eq. (9) is introduced to excite a 4-nm argon gas spot to emit photocurrent radiation $E_\text {r}$, namely the temporal profile depicted in Fig. 1(a), and then the emitted photocurrent radiation propagates towards both sides. Second, the photocurrent radiation propagates towards left is absorbed by the Absorbing Boundary Conditions (ABCs) [34], whereas the counterpart that propagates towards right will meet the pre-placed 23.5-$\mu$m uniform plasma with free electron density $\rho _\text {e} = 2.08\times 10^{24}$m$^{-3}$ (corresponding to plasma frequency $\nu _\text {p} = 13$ THz) and interact with it. When the photocurrent radiation interacts with the uniform plasma, the components above frequency $\nu _\text {p}$ mainly experience dispersion and negligible decaying. However, the components below frequency $\nu _\text {p}$ would be significantly decayed due to the plasma cut-off frequency, and the lower the frequency, the greater the decaying. Based on the decaying and dispersion effect, the photocurrent radiation would be totally modulated by plasma. Third, after experiencing plasma-induced modulation, one part of the modulated photocurrent radiation will propagate through the plasma to the right boundary and be absorbed by the ABCs, whereas the another part will propagate back towards left and is observed at point $P_{\text {BW}}$. It should be noted that, because this case is based on the hypothesis that the photocurrent radiation affects the terahertz generation via plasma-induced modulation, the employed medium is an uniform plasma rather than argon gas. Through resolving the Eqs. (5–7) using the above hypothesis and setting $\rho _\text {e} = 2.08\times 10^{24}$m$^{-3}$, the outcome of backward terahertz emission is depicted by the black-solid line in Fig. 5. Notably, the resemblance between this result and the black-solid line in Fig. 2 affirms the correctness of the interpretation that the photocurrent radiation affects the terahertz generation via plasma-induced modulation.

Through the above analyses, we further introduce a comprehensive three-step diagram to explicate the entire process of terahertz generation from plasma filament, visually depicted in Fig. 6: 1) two-color field excites and ionizes the argon gas, yielding photocurrent radiation at each space point; 2) the photocurrent radiations generated at different points are modulated by plasma in different degrees; 3) the modulated photocurrent radiations are superposed together via interference effect to generate final terahertz emissions in both directions.

 

Fig. 6. The three-step diagram that illustrates the physical process of terahertz generation: 1) the two-color field ionizes argon gas and generates photocurrent radiation; 2) the photocurrent radiation is modulated by plasma; 3) the modulated photocurrent radiations are superposed together and generate terahertz emissions in both directions.

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7. Mechanism verification

In order to substantiate the validity of the above proposed physical mechanism, we apply it to theoretically predict the minimal filament length required for obtaining stable backward terahertz emission, and then conduct simulations to determine the consistency between the theoretical prediction and numerical simulations. The alignment between them would provide a powerful confirmation for the mechanism of terahertz generation—plasma-induced modulation to photocurrent radiation.

In the following analysis, we proceed with the assumption that the permittivity of plasma is characterized by Drude model, because the essence of Eq. (2) and Eq. (7) is a Drude model. Thus, the permittivity is formulated as $\varepsilon _\text {r} = 1- \Delta \varepsilon _\text {p}/\omega ^2$, where $\Delta \varepsilon _\text {p} = e^2\rho _\text {e}/(m_\text {e}\varepsilon _0)$ with $\rho _\text {e} = 2.08\times 10^{24}$ m$^{-3}$. Moreover, the plasma frequency is $\nu _\text {p} = \sqrt {\Delta \varepsilon _\text {p}}/2\pi \approx 13$ THz and the plasma wavelength is $\lambda _\text {p} = c/\nu _\text {p} \approx 23$ $\mu$m. In addition, when the frequency of electric field satisfies to $\nu < \nu _\text {p}$, $\varepsilon _\text {r}$ is a negative value, which yields a purely imaginary refractive index $n = \sqrt {\varepsilon _\text {r}} = i \sqrt {-\varepsilon _\text {r}}$. Therefore, the propagating plane wave $E_\text {x}(z) = E_{\text {x0}}\, \text {exp}(iknz)$ reduces to evanescent wave. When the wave magnitude diminishes to $\text {exp}(-2.3) \approx 0.1$, the contribution of electric field to terahertz emission becomes negligible. Based on this, we can define the minimal filament length as $L_\text {c} = 2.3/k\sqrt {-\varepsilon _\text {r}}$ according to the condition of $kz\sqrt {-\varepsilon _\text {r}} = 2.3$. By substituting the values of $k$ and $\varepsilon _\text {r}$ into the definition of $L_\text {c}$, it turns to

In order to rigorously check the accuracy of the theoretically-predicted $L_\text {c}$, we provide two representative simulations. The first one is to simulate the process that photocurrent radiation $E_\text {r}$ of Fig. 1(a) is modulated by uniform plasma (free electron density $\rho _\text {e} = 2.08 \times 10^{24}$ m$^{-3}$) in different length. The photocurrent radiation $E_\text {r}$ is generated by two-color field $E_{\text {in}}$ given by Eq. (9) exciting a 4-nm argon gas spot. The second one is to simulate an actual backward terahertz emission by using two-color field $E_{\text {in}}$ to excite argon gas layer (atomic density $\rho _{\text {at}} = 2.7\times 10^{25}$ m$^{-3}$) in different length. In addition, the thickness of the two media ranges from 4 nm to 40 $\mu$m during the simulations.

Figure 7 presents the simulation results for the two examples. The outcomes unequivocally demonstrate that, as the length of plasma and argon gas increases, the backward spectrum gradually broadens until the media reach a certain length. In addition, the white straight line corresponds to the theoretically-predicted minimal filament length $L_\text {c} = 11.7\,\mu$m.

 

Fig. 7. The backward terahertz spectra versus medium length and frequency, where the spectral magnitude is normalized by maximal intensity: (a) the backward spectrum of photocurrent radiation $E_\text {r}$ modulated by uniform plasma; (b) the backward spectrum of terahertz emission generated by two-color field exciting argon gas layer. The white straight line represents the minimal filament length $L_\text {c} = 11.7$ $\mu$m theoretically-predicted by Eq. (11).

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In Fig. 7(a), the plasma-induced modulation to the photocurrent radiation leads to absorption of frequency components below $\nu _\text {p} = 13$ THz, exhibiting obvious attenuation in different degrees. Particularly, the low-frequency components experience a precipitous decline, notably that the segments under 0.2 THz that almost diminishes entirely. In contrast, the spectral band from 0.2 THz to 13 THz illustrates relatively weak absorption. Furthermore, as the plasma length increases, the spectrum initially undergoes broadening and then the broadening effect gradually reaches saturation. As shown, once the plasma length surpasses the theoretically-predicted minimal length $L_\text {c}$, the backward radiation stabilizes and undergoes no further alteration. Thus, this alignment between theoretical prediction and numerical simulation suggests the correctness of the minimal filament length.

From Fig. 7(b), it is evident that backward terahertz emission generated from two-color field $E_{\text {in}}$ exciting argon gas emerges around 2-$\mu$m argon gas, and its spectrum predominantly concentrates at zero frequency point. As the argon gas length gradually increases from 2 $\mu$m to the minimal filament length $L_\text {c}$, the spectrum broadens progressively. When reaching and surpassing the predicted length $L_\text {c}$, a stable backward terahertz spectrum is obtained. What’s more, in contrast to Fig. 7(a), the spectrum near the zero-frequency point does not decay to zero but maintain on a stable value, which is attributed to the multi-points excitation of low-frequency components during this process. Therefore, the findings drawn in Fig. 7 (b) are also in harmony with the theoretical prediction for the minimal filament length.

Based upon the above simulation and analysis, we see that the theoretical results rooted in Eq. (11) are effectively substantiated, which means the mechanism of plasma-induced modulation to photocurrent radiation does meet the expectation of theoretical prediction and numerical simulation. Therefore, it is convincingly confirmed that plasma-induced modulation to photocurrent radiation does be in dominance within the process of terahertz generation from two-color laser induced plasma filament.

8. Conclusion

By rigorously analyzing terahertz generation from two-color laser-induced plasma filament based on photocurrent model, we successfully explain the respective roles of pump field and photocurrent radiation in terahertz generation. By comparing the numerical simulation results under situations of considering and disregarding photocurrent radiation respectively, the contrasting results clearly prove that the driving effect of photocurrent radiation has a significant impact on terahertz generation from laser-induced plasma filament. Furthermore, we conduct two numerical experiments to further investigate the driving effect of photocurrent radiation and confirm that the underlying physical mechanism of terahertz generation is plasma-induced modulation to photocurrent radiation. Moreover, based on this conclusion, this work proposes a three-step diagram to illustrate the entire physical process in detail. Finally, the correctness of this model is definitively verified through theoretically predicting the minimal filament length required to sustain stable backward terahertz emission based on the mechanism of plasma-induced modulation to photocurrent radiation. The consistency between theoretical prediction and numerical simulation confirms the credibility of the mechanism and the proposed three-step diagram.